This all looks quite hard to follow. Let me abbreviate a bit more. Let's write "$P$" for "$1$", and "$Q$" for "$2$" and "$\wedge$" for "$0$". (Not that this makes any difference. Strings are just strings. They have no "intrinsic" meaning.) Furthermore, instead of

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no. σ1 is also a logical consequence of σ2 if counting and precedes that event. Numeric representation of 1 is 1 and 2 is 11 as the basis for the numeral two. In sequence it would be more accurate to include how numerals are established via counting. It is a logical operation on the assumption of truth of the sequence as a fixed event versus constructed. Thus one establishes two before it stands on its own. The existing approach is still valid yet relative and secondary. The question is multifaceted.